Piston rings are installed in the piston and cylinder wall, which does not have a perfect round shape due to machining tolerances or external loads e.g. head bolts tightening. If the ring cannot follow these deformations, a localized lack of contact will occur and consequently an increase in the engine blow-by and lubricant oil consumption. Current 2D computational methods can not implement such effects – more complex model is necessary. The presented master’s thesis is focused on the developement of a flexible 3D piston ring model able to capture local deformations. It is based on the Timoshenko beam theory in cooperation with MBS software Adams. Model is then compared with FEM using software ANSYS. The validated piston ring model is assembled into the piston/cylinder liner and very basic simulations are run. Finally, future improvements are suggested.

Dans ce travail, différentes formulations d'éléments de poutres sont proposées pour l'analyse à rupture de structures de type poutres ou portiques en béton armé soumises à des chargements statiques monotones. La rupture localisée des matériaux est modélisée par la méthode à discontinuité forte, qui consiste à enrichir l'interpolation standard des déplacements (ou rotations) avec des fonctions discontinues associées à un paramètre cinématique supplémentaire interprété comme un saut de déplacement (ou rotation). Ces paramètres additionnels sont locaux et condensés au niveau élémentaire. Un élément fini écrit en efforts résultants et deux éléments finis multi-couches sont développés dans ce travail. L'élément de poutre d'Euler Bernouilli écrit en effort résultant présente une discontinuité en rotation. La réponse en flexion du matériau hors discontinuité est décrite par un modèle élastoplastique en effort résultant et la relation cohésive liant moment et saut de rotation sur la rotule plastique est, quant à elle, décrite par un modèle rigide plastique. La réponse axiale est suppposée élastique. Pour ce qui concerne l'approche multi-couche, chaque couche est considérée comme une barre constituée de béton ou d'acier. La partie régulière de la déformation de chaque couche est calculée en s'appuyant sur la cinématique associée à la théorie d'Euler Bernoulli ou de Timoshenko. Une déformation axiale additionnelle est considérée par l'introduction d'une discontinuité du déplacement axial, introduite indépendamment dans chaque couche. Le comportement du béton est pris en compte par un modèle élasto-endommageable alors que celui de l'acier est décrit par un modèle élastoplastique. La relation cohésive entre la traction sur la discontinuité et le saut de déplacement axial est décrit par un modèle rigide endommageable adoucissant pour les barres (couches) en béton et rigide plastique adoucissant pour les barres en acier. La réponse en cisaillement pour l'élement de Timoshenko est supposée élastique. Enfin, l'élément multi-couche de Timoshenko est enrichi en introduisant une partie visqueuse dans la réponse adoucissante. L'implantation numérique des différents éléments développés dans ce travail est présentée en détail. La résolution par une procédure d'«operator split» est décrite pour chaque type d'élément. Les différentes quantités nécessaires pour le calcul au niveau local des variables internes des modèles non linéaires ainsi que pour la construction du système global fournissant les valeurs des dégrés de liberté sont précisées. Les performances des éléments développés sont illustrées à travers des exemples numériques montrant que la formulation basée sur un élément multicouche d'Euler Bernouilli n'est pas robuste alors les simulations s'appuyant sur des éléments d'Euler Bernouilli en efforts résultants ou sur des éléments multicouche de Timoshenko fournissent des résultats très satisfaisants. / In this work, several beam finite element formulations are proposed for failure analysis of planar reinforced concrete beams and frames under monotonic static loading. The localized failure of material is modeled by the embedded strong discontinuity concept, which enhances standard interpolation of displacement (or rotation) with a discontinuous function, associated with an additional kinematic parameter representing jump in displacement (or rotation). The new parameters are local and are condensed on the element level. One stress resultant and two multi-layer beam finite elements are derived. The stress resultant Euler-Bernoulli beam element has embedded discontinuity in rotation. Bending response of the bulk of the element is described by elasto-plastic stress resultant material model. The cohesive relation between the moment and the rotational jump at the softening hinge is described by rigid-plastic model. Axial response is elastic. In the multi-layer beam finite elements, each layer is treated as a bar, made of either concrete or steel. Regular axial strain in a layer is computed according to Euler-Bernoulli or Timoshenko beam theory. Additional axial strain is produced by embedded discontinuity in axial displacement, introduced individually in each layer. Behavior of concrete bars is described by elastodamage model, while elasto-plasticity model is used for steel bars. The cohesive relation between the stress at the discontinuity and the axial displacement jump is described by rigid-damage softening model in concrete bars and by rigid-plastic softening model in steel bars. Shear response in the Timoshenko element is elastic. Finally, the multi-layer Timoshenko beam finite element is upgraded by including viscosity in the softening model. Computer code implementation is presented in detail for the derived elements. An operator split computational procedure is presented for each formulation. The expressions, required for the local computation of inelastic internal variables and for the global computation of the degrees of freedom, are provided. Performance of the derived elements is illustrated on a set of numerical examples, which show that the multi-layer Euler-Bernoulli beam finite element is not reliable, while the stress-resultant Euler-Bernoulli beam and the multi-layer Timoshenko beam finite elements deliver satisfying results. / V disertaciji predlagamo nekaj formulacij končnih elementov za porušno analizo armiranobetonskih nosilcev in okvirjev pod monotono statično obteˇzbo. Lokalizirano porušitev materiala modeliramo z metodo vgrajene nezveznosti, pri kateri standardno interpolacijo pomikov (ali zasukov) nadgradimo z nezvezno interpolacijsko funkcijo in z dodatnim kinematičnim parametrom, ki predstavlja velikost nezveznosti v pomikih (ali zasukih). Dodatni parametri so lokalnega značaja in jih kondenziramo na nivoju elementa. Izpeljemo en rezultantni in dva večslojna končna elementa za nosilec. Rezultantni element za Euler-Bernoullijev nosilec ima vgrajeno nezveznost v zasukih. Njegov upogibni odziv opišemo z elasto-plastičnim rezultantnim materialnim modelom. Kohezivni zakon, ki povezuje moment v plastičnem členku s skokom v zasuku, opišemo s togo-plastičnim modelom mehčanja. Osni odziv je elastičen. V večslojnih končnih elementih vsak sloj obravnavamo kot betonsko ali jekleno palico. Standardno osno deformacijo v palici izračunamo v skladu z Euler-Bernoullijevo ali s Timošenkovo teorijo nosilcev. Vgrajena nezveznost v osnem pomiku povzroči dodatno osno deformacijo v posamezni palici. Obnašanje betonskega sloja opišemo z modelom elasto-poškodovanosti, za sloj armature pa uporabimo elasto-plastični model. Kohezivni zakon, ki povezuje napetost v nezveznosti s skokom v osnem pomiku, opišemo z modelom mehčanja v poškodovanosti za beton in s plastičnim modelom mehčanja za jeklo.Striˇzni odziv Timošenkovega nosilca je elastičen. Večslojni končni element za Timošenkov nosilec nadgradimo z viskoznim modelom mehčanja. Za vsak končni element predstavimo računski algoritem ter vse potrebne izraze za lokalni izračun neelastičnih notranjih spremenljivk in za globalni izračun prostostnih stopenj. Delovanje končnih elementov preizkusimo na več numeričnih primerih. Ugotovimo, da večslojni končni element za Euler-Bernoullijev nosilec ni zanesljiv, medtem ko rezultantni končni element za Euler-Bernoullijev nosilec in večslojni končni element za Timošenkov nosilec dajeta zadovoljive rezultate.

In der vorliegenden Arbeit werden ein Berechnungs- und Sicherheitskonzept für Verbundbauteile in Holzverbundbauweise erarbeitet und validiert. Damit ist eine globale Dimensionierung bzw. Bemessung dieser Bauteile möglich. Der Fokus der Konzepte liegt dabei auf Anwendungen und Anforderungen im Maschinenbau. Hierzu werden zunächst Grundlagen zu Holzwerkstoffen und dem Lastfall Dreipunktbiegung erarbeitet. Darauf aufbauend werden in Dreipunktbiegeversuchen die Biege- und Schubeigenschaften ausgewählter Sperrhölzer ermittelt. Im nächsten Schritt wird ein Berechnungskonzept zur Vorhersage der Kraft-Verformungs-Kurve von Bauteilen in Holzbauweise unter Verwendung der Materialkennwerte der Einzelelemente erarbeitet und validiert. Die Validierung erfolgt anhand ausgewählter Versuchsmuster in Holzbauweise. Anschließend wird ein an die Sicherheitsanforderungen im Maschinenbau angepasstes semiprobabilistisches Sicherheitskonzept erarbeitet. Abschließend werden anhand praxisnaher Beispiele die Anwendbarkeit der Konzepte validiert und Möglichkeiten bzw. Grenzen aufgezeigt.:1 Einleitung 14 1.1 Motivation 14 1.2 Präzisierung der Aufgabenstellung 16 1.3 Lösungsansätze und Abgrenzung der Arbeit 17 1.4 Aufbau der Arbeit 18 2 Grundlagen 20 2.1 Ingenieurtechnische Grundlagen 20 2.1.1 Begriffsdefinition 20 2.1.2 Zusammenhang zwischen Kraft- und Verformungsgrößen 21 2.1.3 Materialkennwerte 23 2.1.4 Verbundbauteile 24 2.2 Grundlagen der Dreipunktbiegung 25 2.2.1 Allgemeine Modellannahmen 27 2.2.2 Timoshenko-Balkentheorie 27 2.2.3 Einfluss des Stützweiten-Höhen-Verhältnisses 30 2.2.4 Schubkorrekturfaktor 32 2.3 Grundlagen zum Werkstoff Holz 33 2.3.1 Struktureller Aufbau 33 2.3.2 Inhomogenität von Holz 35 2.3.3 Anisotropie des Holzes 35 2.3.4 Mechanische Eigenschaften 39 2.3.5 Äußere Einflussfaktoren auf die Materialeigenschaften von Holz 41 2.4 Holzfurnierlagenverbundwerkstoffe (WVC) 42 2.4.1 Einteilung 42 2.4.2 Struktureller Aufbau von Sperrholz 44 2.4.3 Mechanische Eigenschaften von Sperrholz 44 2.4.4 Spannungszustand von Sperrholz bei Dreipunktbiegebeanspruchung 46 3 Materialcharakterisierung 49 3.1 Stand der Technik 49 3.1.1 Literaturkennwerte, Materialdatenblätter und Leistungserklärungen 49 3.1.2 Aktuelle Normung 49 3.1.3 Kritik am Stand der Technik 51 3.1.4 Ableitung von Anforderungen an Materialversuche und Kennwerte 53 3.1.5 Ziele der Materialcharakterisierung 54 3.2 Grundlagen der Datenanalyse 55 3.2.1 Statistische Lage- und Streumaße 55 3.2.2 Graphische Darstellung empirischer Daten 56 3.2.3 Normalverteilung 57 3.2.4 Statistische Testverfahren 58 3.3 Material und Methoden 60 3.3.1 Material 60 3.3.2 Stützweitenversuch – Versuchssetup und Auswertemethodik 62 3.3.3 Kurzbiegeversuche – Versuchssetup und Auswertemethodik 67 3.4 Kennwertermittlung 68 3.4.1 Materialcharakterisierung von WVC-01 68 3.4.2 Materialcharakterisierung von WVC-02 75 3.4.3 Universelle Spannungs-Dehnungs-Kennwerte 77 4 Berechnungskonzept für Verbundbauteile in Holzbauweise 82 4.1 Annahmen und Eingrenzung 82 4.2 Aktueller Stand der Technik 82 4.2.1 Berechnung der Bauteilsteifigkeit von Verbundbauteilen 82 4.2.2 Berechnung von Versagenspunkten 83 4.2.3 Kritik am Stand der Technik 86 4.2.4 Zielstellung 88 4.3 Berechnungskonzept für Verbundbauteile aus Holzwerkstoffen 88 4.3.1 Prinzipielles Vorgehen 88 4.3.2 Berechnung der Bauteilmodulkennwerte 89 4.3.3 Berechnung der Versagenspunkte 91 4.3.4 Berechnung der Geometrieparameter bei gegebener Mindesttragfähigkeit 92 4.4 Validierung des Berechnungskonzeptes 94 4.4.1 Methodik 94 4.4.2 Aufbau und Geometrie 94 4.4.3 Experimentelle Ergebnisse der Bauteiltests 99 4.4.4 Berechnung der Tragfähigkeit mit Materialkennwerten 100 4.4.5 Berechnung mit universellen Normaldehnungen 105 4.4.6 Diskussion 106 5 Sicherheitskonzept für Holzwerkstoffe im Maschinenbau 108 5.1 Stand der Technik 108 5.1.1 Sicherheit – Definition und Arten 108 5.1.2 Sicherheit im Maschinenbau 113 5.1.3 Sicherheit in der Kunststofftechnik 113 5.1.4 Sicherheit im Ingenieurholzbau – EUROCODE 5 114 5.1.5 Kritik am Stand der Technik 117 5.1.6 Ziel des Sicherheitskonzeptes für Holzwerkstoffe im Maschinenbau 119 5.2 Entwicklung des Sicherheitskonzeptes 120 5.2.1 Analyse der Teilsicherheitsbeiwerte des EUROCODE 5 120 5.2.2 Teilsicherheitsbeiwerte für statische Lastfälle des Maschinenbau 122 5.2.3 Beiwert zur Berücksichtigung der Kennwertstreuung 125 5.2.4 Sicherheitsbezogene Klassifizierung von Maschinenbauanwendungen 126 5.2.5 Ableitung globaler Sicherheitsfaktoren 129 5.2.6 Zusammenfassung 132 5.3 Validierung 133 5.3.1 Bemessung nach EUROCODE 5 133 5.3.2 Bemessung nach Sicherheitskonzept für Maschinenbau 135 5.3.3 Vergleich EUROCODE 5 und Sicherheitskonzept für Maschinenbau 136 6 Anwendbarkeitsstudie 139 6.1 Bemessung von Hohlprofilen 139 6.2 Dimensionierung von Hohlprofilen 142 6.3 Globale Bemessung komplexer Bauteile 144 6.4 Anschließende Bemessungsaufgaben 147 7 Zusammenfassung und Ausblick 149 7.1 Zusammenfassung 149 7.2 Ausblick 151 8 Verzeichnisse 152 9 Anhang 169 9.1 Anhang zu Kapitel 2 169 9.2 Anhang zu Kapitel 3 171 9.3 Anhang zu Kapitel 4 194 9.4 Anhang zu Kapitel 5 207 9.5 Anhang zu Kapitel 6 212 / In the present work, a calculation and safety concept for composite components in wood composite construction is developed and validated. This enables a global dimensioning of these components. The focus of the concepts is on applications and requirements in mechanical engineering. For this purpose, the basics of wood-based materials and the three-point bending load case are first elaborated. Based on this, the bending and shear properties of selected plywood are determined in three-point bending tests. In the next step, a calculation concept for predicting the force-deformation-curve of components in timber construction using the material parameters of the individual elements will be developed and validated. The validation is based on selected test components. A semi-probabilistic safety concept adapted to the safety requirements in mechanical engineering is then developed. Finally, using practical examples, the applicability of the concepts is determined and possibilities and limits are shown.:1 Einleitung 14 1.1 Motivation 14 1.2 Präzisierung der Aufgabenstellung 16 1.3 Lösungsansätze und Abgrenzung der Arbeit 17 1.4 Aufbau der Arbeit 18 2 Grundlagen 20 2.1 Ingenieurtechnische Grundlagen 20 2.1.1 Begriffsdefinition 20 2.1.2 Zusammenhang zwischen Kraft- und Verformungsgrößen 21 2.1.3 Materialkennwerte 23 2.1.4 Verbundbauteile 24 2.2 Grundlagen der Dreipunktbiegung 25 2.2.1 Allgemeine Modellannahmen 27 2.2.2 Timoshenko-Balkentheorie 27 2.2.3 Einfluss des Stützweiten-Höhen-Verhältnisses 30 2.2.4 Schubkorrekturfaktor 32 2.3 Grundlagen zum Werkstoff Holz 33 2.3.1 Struktureller Aufbau 33 2.3.2 Inhomogenität von Holz 35 2.3.3 Anisotropie des Holzes 35 2.3.4 Mechanische Eigenschaften 39 2.3.5 Äußere Einflussfaktoren auf die Materialeigenschaften von Holz 41 2.4 Holzfurnierlagenverbundwerkstoffe (WVC) 42 2.4.1 Einteilung 42 2.4.2 Struktureller Aufbau von Sperrholz 44 2.4.3 Mechanische Eigenschaften von Sperrholz 44 2.4.4 Spannungszustand von Sperrholz bei Dreipunktbiegebeanspruchung 46 3 Materialcharakterisierung 49 3.1 Stand der Technik 49 3.1.1 Literaturkennwerte, Materialdatenblätter und Leistungserklärungen 49 3.1.2 Aktuelle Normung 49 3.1.3 Kritik am Stand der Technik 51 3.1.4 Ableitung von Anforderungen an Materialversuche und Kennwerte 53 3.1.5 Ziele der Materialcharakterisierung 54 3.2 Grundlagen der Datenanalyse 55 3.2.1 Statistische Lage- und Streumaße 55 3.2.2 Graphische Darstellung empirischer Daten 56 3.2.3 Normalverteilung 57 3.2.4 Statistische Testverfahren 58 3.3 Material und Methoden 60 3.3.1 Material 60 3.3.2 Stützweitenversuch – Versuchssetup und Auswertemethodik 62 3.3.3 Kurzbiegeversuche – Versuchssetup und Auswertemethodik 67 3.4 Kennwertermittlung 68 3.4.1 Materialcharakterisierung von WVC-01 68 3.4.2 Materialcharakterisierung von WVC-02 75 3.4.3 Universelle Spannungs-Dehnungs-Kennwerte 77 4 Berechnungskonzept für Verbundbauteile in Holzbauweise 82 4.1 Annahmen und Eingrenzung 82 4.2 Aktueller Stand der Technik 82 4.2.1 Berechnung der Bauteilsteifigkeit von Verbundbauteilen 82 4.2.2 Berechnung von Versagenspunkten 83 4.2.3 Kritik am Stand der Technik 86 4.2.4 Zielstellung 88 4.3 Berechnungskonzept für Verbundbauteile aus Holzwerkstoffen 88 4.3.1 Prinzipielles Vorgehen 88 4.3.2 Berechnung der Bauteilmodulkennwerte 89 4.3.3 Berechnung der Versagenspunkte 91 4.3.4 Berechnung der Geometrieparameter bei gegebener Mindesttragfähigkeit 92 4.4 Validierung des Berechnungskonzeptes 94 4.4.1 Methodik 94 4.4.2 Aufbau und Geometrie 94 4.4.3 Experimentelle Ergebnisse der Bauteiltests 99 4.4.4 Berechnung der Tragfähigkeit mit Materialkennwerten 100 4.4.5 Berechnung mit universellen Normaldehnungen 105 4.4.6 Diskussion 106 5 Sicherheitskonzept für Holzwerkstoffe im Maschinenbau 108 5.1 Stand der Technik 108 5.1.1 Sicherheit – Definition und Arten 108 5.1.2 Sicherheit im Maschinenbau 113 5.1.3 Sicherheit in der Kunststofftechnik 113 5.1.4 Sicherheit im Ingenieurholzbau – EUROCODE 5 114 5.1.5 Kritik am Stand der Technik 117 5.1.6 Ziel des Sicherheitskonzeptes für Holzwerkstoffe im Maschinenbau 119 5.2 Entwicklung des Sicherheitskonzeptes 120 5.2.1 Analyse der Teilsicherheitsbeiwerte des EUROCODE 5 120 5.2.2 Teilsicherheitsbeiwerte für statische Lastfälle des Maschinenbau 122 5.2.3 Beiwert zur Berücksichtigung der Kennwertstreuung 125 5.2.4 Sicherheitsbezogene Klassifizierung von Maschinenbauanwendungen 126 5.2.5 Ableitung globaler Sicherheitsfaktoren 129 5.2.6 Zusammenfassung 132 5.3 Validierung 133 5.3.1 Bemessung nach EUROCODE 5 133 5.3.2 Bemessung nach Sicherheitskonzept für Maschinenbau 135 5.3.3 Vergleich EUROCODE 5 und Sicherheitskonzept für Maschinenbau 136 6 Anwendbarkeitsstudie 139 6.1 Bemessung von Hohlprofilen 139 6.2 Dimensionierung von Hohlprofilen 142 6.3 Globale Bemessung komplexer Bauteile 144 6.4 Anschließende Bemessungsaufgaben 147 7 Zusammenfassung und Ausblick 149 7.1 Zusammenfassung 149 7.2 Ausblick 151 8 Verzeichnisse 152 9 Anhang 169 9.1 Anhang zu Kapitel 2 169 9.2 Anhang zu Kapitel 3 171 9.3 Anhang zu Kapitel 4 194 9.4 Anhang zu Kapitel 5 207 9.5 Anhang zu Kapitel 6 212

The thesis presents a novel computational method for analysing the static and dynamic behaviour of a multi-damaged beam using local and non-local elasticity theories. Most of the lumped damage beam models proposed to date are based on slender beam theory in classical (local) elasticity and are limited by inaccuracies caused by the implicit assumption of the Euler-Bernoulli beam model and by the spring model itself, which simplifies the real beam behaviour around the crack. In addition, size effects and material heterogeneity cannot be taken into account using the classical elasticity theory due to the absence of any microstructural parameter. The proposed work is based on the inhomogeneous Euler-Bernoulli beam theory in which a Dirac's delta function is added to the bending flexibility at the position of each crack: that is, the severer the damage, the larger is the resulting impulsive term. The crack is assumed to be always open, resulting in a linear system (i.e. nonlinear phenomena associated with breathing cracks are not considered). In order to provide an accurate representation of the structure's behaviour, a new multi-cracked beam element including shear effects and rotatory inertia is developed using the flexibility approach for the concentrated damage. The resulting stiffness matrix and load vector terms are evaluated by the unit-displacement method, employing the closed-form solutions for the multi-cracked beam problem. The same deformed shapes are used to derive the consistent mass matrix, also including the rotatory inertia terms. The two-node multi-damaged beam model has been validated through comparison of the results of static and dynamic analyses for two numerical examples against those provided by a commercial finite element code. The proposed model is shown to improve the computational efficiency as well as the accuracy, thanks to the inclusion of both shear deformations and rotatory inertia. The inaccuracy of the spring model, where for example for a rotational spring a finite jump appears on the rotations' profile, has been tackled by the enrichment of the elastic constitutive law with higher order stress and strain gradients. In particular, a new phenomenological approach based upon a convenient form of non-local elasticity beam theory has been presented. This hybrid non-local beam model is able to take into account the distortion on the stress/strain field around the crack as well as to include the microstructure of the material, without introducing any additional crack related parameters. The Laplace's transform method applied to the differential equation of the problem allowed deriving the static closed-form solution for the multi-cracked Euler-Bernoulli beams with hybrid non-local elasticity. The dynamic analysis has been performed using a new computational meshless method, where the equation of motions are discretised by a Galerkin-type approximation, with convenient shape functions able to ensure the same grade of approximation as the beam element for the classical elasticity. The importance of the inclusion of microstructural parameters is addressed and their effects are quantified also in comparison with those obtained using the classical elasticity theory.

Este trabalho fornece uma contribuição à análise estrutural não-linear de pavimentos de edifícios de concreto armado com o emprego do Método dos Elementos Finitos. A deformação por esforço cortante é considerada, portanto as teorias de Timoshenko e de Reissner-Mindlin são empregadas nas formulações dos elementos de barra e de placa, respectivamente. As posições dos elementos de barra e de placa são independentes e, portanto, podem ser definidas em diferentes planos. Em conseqüência do exposto, o efeito de membrana deve ser necessariamente considerado na modelagem do pavimento. Para completar o modelo mecânico, as não-linearidades físicas descrevem o comportamento do concreto e do aço. A deterioração do concreto no cisalhamento é também considerada através de um modelo simplificado que é proposto para a modelagem do cisalhamento em condições de serviço. / This work gives a contribution to the non-linear structural analysis of reinforced concrete buildings floors using the Finite Element Method. The shear strain components are taken into account by adopting the Timoshenko\'s beam theory together with and the Reissner-Mindlin\'s theory for plate bending. Bar and plate element position are independent and therefore can be defined at different planes. As several level are considered when defining the structure membrane effects are necessary considered. In order to complete the mechanical model, physical non-linearities are also assumed to describe concrete and steel behaviours. The deterioration of the concrete material in shear is also taken account. For this purpose, a simplified model is adopted to compute approximately the damaged shear component in the steel direction.

This work was supported by Fundação para a Ciência e a Tecnologia, through the scholarship SFRH/BD/35821/2007 / Tese de doutoramento. Engenharia Mecânica. Faculdade de Engenharia. Universidade do Porto. 2012

This thesis presents an approach to modelling dynamic mass capture which is applied to a number of system models. The models range from a simple 2D Euler-Bernoulli beam with point masses for the end-effector and target to a 3D Timoshenko beam model (including torsion) with rigid bodies for the end-effector and target. In addition, new models for torsion, as well as software to derive the finite element equations from first principles were developed to support the modelling. Results of the models are compared to a simple experiment as done by Ben Rhody. Investigations of offset capture are done by simulation to show why one would consider using a 3D model that includes torsion. These problems have relevance to both terrestrial robots and to space based robotic systems such as the manipulators on the International Space Station capturing payloads such as the SpaceX Dragon capsule. One could increase production in an industrial environment if industrial robots could pick up items without having to establish a zero relative velocity between the end effector and the item. To have a robot acquire its payload in this way would introduce system dynamics that could lead to the necessity of modelling a previously ‘rigid’ robot as flexible.

This thesis is about using structural-dynamics based methods to address the existing challenges in the field of Structural Health Monitoring (SHM). Particularly, new structural-dynamics based methods are presented, to model areas of damage, to do damage diagnosis and to estimate and predict the sensitivity of structural vibration properties like natural frequencies to the presence of damage. Towards these objectives, a general analytical procedure, which yields nth-order expressions governing mode shapes and natural frequencies and for damaged elastic structures such as rods, beams, plates and shells of any shape is presented. Features of the procedure include the following: 1. Rather than modeling the damage as a fictitious elastic element or localized or global change in constitutive properties, it is modeled in a mathematically rigorous manner as a geometric discontinuity. 2. The inertia effect (kinetic energy), which, unlike the stiffness effect (strain energy), of the damage has been neglected by researchers, is included in it. 3. The framework is generic and is applicable to wide variety of engineering structures of different shapes with arbitrary boundary conditions which constitute self adjoint systems and also to a wide variety of damage profiles and even multiple areas of damage. To illustrate the ability of the procedure to effectively model the damage, it is applied to beams using Euler-Bernoulli and Timoshenko theories and to plates using Kirchhoff's theory, supported on different types of boundary conditions. Analytical results are compared with experiments using piezoelectric actuators and non-contact Laser-Doppler Vibrometer sensors. Next, the step of damage diagnosis is approached. Damage diagnosis is done using two methodologies. One, the modes and natural frequencies that are determined are used to formulate analytical expressions for a strain energy based damage index. Two, a new damage detection parameter are identified. Assuming the damaged structure to be a linear system, the response is expressed as the summation of the responses of the corresponding undamaged structure and the response (negative response) of the damage alone. If the second part of the response is isolated, it forms what can be regarded as the damage signature. The damage signature gives a clear indication of the damage. In this thesis, the existence of the damage signature is investigated when the damaged structure is excited at one of its natural frequencies and therefore it is called ``partial mode contribution". The second damage detection method is based on this new physical parameter as determined using the partial mode contribution. The physical reasoning is verified analytically, thereupon it is verified using finite element models and experiments. The limits of damage size that can be determined using the method are also investigated. There is no requirement of having a baseline data with this damage detection method. Since the partial mode contribution is a local parameter, it is thus very sensitive to the presence of damage. The parameter is also shown to be not affected by noise in the detection ambience.

Uncertainties in load and system properties play a significant role in reliability analysis of vibrating structural systems. The subject of random vibrations has evolved over the last few decades to deal with uncertainties in external loads. A well developed body of literature now exists which documents the status of this subject. Studies on the influ­ence of system property uncertainties on reliability of vibrating structures is, however, of more recent origin. Currently, the problem of dynamic response characterization of sys­tems with parameter uncertainties has emerged as a subject of intensive research. The motivation for this research activity arises from the need for a more accurate assess­ment of the safety of important and high cost structures like nuclear plant installations, satellites and long span bridges. The importance of the problem also lies in understand­ing phenomena like mode localization in nearly periodic structures and deviant system behaviour at high frequencies. It is now well established that these phenomena are strongly influenced by spatial imperfections in the vibrating systems. Design codes, as of now, are unable to systematically address the influence of scatter and uncertainties. Therefore, there is a need to develop robust design algorithms based on the probabilistic description of the uncertainties, leading to safer, better and less over-killed designs. Analysis of structures with parameter uncertainties is wrought with diffi­culties, which primarily arise because the response variables are nonlinearly related to the stochastic system parameters; this being true even when structures are idealized to display linear material and deformation characteristics. The problem is further com­pounded when nonlinear structural behaviour is included in the analysis. The analysis of systems with parameter uncertainties involves modeling of random fields for the system parameters, discretization of these random fields, solutions of stochastic differential and algebraic eigenvalue problems, inversion of random matrices and differential operators, and the characterization of random matrix products. It should be noted that the mathematical nature of many of these problems is substantially different from those which are encountered in the traditional random vibration analysis. The basic problem lies in obtaining the solution of partial differential equations with random coefficients which fluctuate in space. This has necessitated the development of methods and tools to deal with these newer class of problems. An example of this development is the generalization of the finite element methods of structural analysis to encompass problems of stochastic material and geometric characteristics. The present thesis contributes to the development of methods and tools to deal with structural uncertainties in the analysis of vibrating structures. This study is a part of an ongoing research program in the Department, which is aimed at gaining insights into the behaviour of randomly parametered dynamical systems and to evolve computational methods to assess the reliability of large scale engineering structures. Recent studies conducted in the department in this direction, have resulted in the for­mulation of the stochastic dynamic stiffness matrix for straight Euler-Bernoulli beam elements and these results have been used to investigate the transient and the harmonic steady state response of simple built-up structures. In the present study, these earlier formulations are extended to derive the stochastic dynamic stiffness matrix for a more general beam element, namely, the curved Timoshenko beam element. Furthermore, the method has also been extended to study the mean and variance of the stationary response of built-up structures when excited by stationary stochastic forces. This thesis is organized into five chapters and four appendices. The first chapter mainly contains a review of the developments in stochas­tic finite element method (SFEM). Also presented is a brief overview of the dynamics of curved beams and the essence of the dynamic stiffness matrix method. This discussion also covers issues pertaining to modeling rotary inertia and shear deformations in the study of curved beam dynamics. In the context of SFEM, suitability of different methods for modeling system uncertainties, depending on the type of problem, is discussed. The relative merits of several schemes of discretizing random fields, namely, local averaging, series expansions using orthogonal functions, weighted integral approach and the use of system Green functions, are highlighted. Many of the discretization schemes reported in the literature have been developed in the context of static problems. The advantages of using the dynamic stiffness matrix approach in conjunction with discretization schemes based on frequency dependent shape functions, are discussed. The review identifies the dynamic analysis of structures built-up of randomly parametered curved beams, using dynamic stiffness matrix method, as a problem requiring further research. The review also highlights the need for studies on the treatment of non-Gaussian nature of system parameters within the framework of stochastic finite element analysis and simulation methods. The problem of deterministic analysis of curved beam elements is consid­ered first. Chapter 2 reports on the development of the dynamic stiffness matrix for a curved Timoshenko beam element. It is shown that when the beam is uniformly param-etered, the governing field equations can be solved in a closed form. These closed form solutions serve as the basis for the formulation of damping and frequency dependent shape functions which are subsequently employed in the thesis to develop the dynamic stiffness matrix of stochastically inhomogeneous, curved beams. On the other hand, when the beam properties vary spatially, the governing equations have spatially varying coefficients which discount the possibility of closed form solutions. A numerical scheme to deal with this problem is proposed. This consists of converting the governing set of boundary value problems into a larger class of equivalent initial value problems. This set of Initial value problems can be solved using numerical schemes to arrive at the element dynamic stiffness matrix. This algorithm forms the basis for Monte Carlo simulation studies on stochastic beams reported later in this thesis. Numerical results illustrating the formulations developed in this chapter are also presented. A satisfactory agreement of these results has been demonstrated with the corresponding results obtained from independent finite element code using normal mode expansions. The formulation of the dynamic stiffness matrix for a curved, randomly in-homogeneous, Timoshenko beam element is considered in Chapter 3. The displacement fields are discretized using the frequency dependent shape functions derived in the pre­vious chapter. These shape functions are defined with respect to a damped, uniformly parametered beam element and hence are deterministic in nature. Lagrange's equations are used to derive the 6x6 stochastic dynamic stiffness matrix of the beam element. In this formulation, the system property random fields are implicitly discretized as a set of damping and frequency dependent Weighted integrals. The results for a straight Timo- shenko beam are obtained as a special case. Numerical examples on structures made up of single curved/straight beam elements are presented. These examples also illustrate the characterization of the steady state response when excitations are modeled as stationary random processes. Issues related to ton-Gaussian features of the system in-homogeneities are also discussed. The analytical results are shown to agree satisfactorily with corresponding results from Monte Carlo simulations using 500 samples. The dynamics of structures built-up of straight and curved random Tim-oshenko beams is studied in Chapter 4. First, the global stochastic dynamic stiffness matrix is assembled. Subsequently, it is inverted for calculating the mean and variance, of the steady state stochastic response of the structure when subjected to stationary random excitations. Neumann's expansion method is adopted for the inversion of the stochastic dynamic stiffness matrix. Questions on the treatment of the beam characteris­tics as non-Gaussian random fields, are addressed. It is shown that the implementation of Neumann's expansion method and Monte-Carlo simulation method place distinc­tive demands on strategy of modeling system parameters. The Neumann's expansion method, on one hand, requires the knowledge of higher order spectra of beam properties so that the non-Gaussian features of beam parameters are reflected in the analysis. On the other hand, simulation based methods require the knowledge of the range of the stochastic variations and details of the probability density functions. The expediency of implementing Gaussian closure approximation in evaluating contributions from higher order terms in the Neumann expansion is discussed. Illustrative numerical examples comparing analytical and Monte-Carlo simulations are presented and the analytical so­lutions are found to agree favourably with the simulation results. This agreement lends credence to the various approximations involved in discretizing the random fields and inverting the global dynamic stiffness matrix. A few pointers as to how the methods developed in the thesis can be used in assessing the reliability of these structures are also given. A brief summary of contributions made in the thesis together with a few suggestions for further research are presented in Chapter 5. Appendix A describes the models of non-Gaussian random fields employed in the numerical examples considered in this thesis. Detailed expressions for the elements of the covariance matrix of the weighted integrals for the numerical example considered in Chapter 5, are presented in Appendix B; A copy of the paper, which has been ac­cepted for publication in the proceedings of IUTAM symposium on 'Nonlinearity and Stochasticity in Structural Mechanics' has been included as Appendix C.

This thesis presents an approach to modelling dynamic mass capture which is applied to a number of system models. The models range from a simple 2D Euler-Bernoulli beam with point masses for the end-effector and target to a 3D Timoshenko beam model (including torsion) with rigid bodies for the end-effector and target. In addition, new models for torsion, as well as software to derive the finite element equations from first principles were developed to support the modelling. Results of the models are compared to a simple experiment as done by Ben Rhody. Investigations of offset capture are done by simulation to show why one would consider using a 3D model that includes torsion. These problems have relevance to both terrestrial robots and to space based robotic systems such as the manipulators on the International Space Station capturing payloads such as the SpaceX Dragon capsule. One could increase production in an industrial environment if industrial robots could pick up items without having to establish a zero relative velocity between the end effector and the item. To have a robot acquire its payload in this way would introduce system dynamics that could lead to the necessity of modelling a previously ‘rigid’ robot as flexible.